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A COUPLED JOINT EIGENVALUE
DECOMPOSITION ALGORITHM FOR CANONICAL
POLYADIC DECOMPOSITION OF TENSORS
Rémi André, Xavier Luciani, Eric Moreau
To cite this version:
Rémi André, Xavier Luciani, Eric Moreau. A COUPLED JOINT EIGENVALUE
DECOMPOSI-TION ALGORITHM FOR CANONICAL POLYADIC DECOMPOSIDECOMPOSI-TION OF TENSORS. IEEE
SAM 2016, Jul 2016, Rio de Janeiro, France. �hal-01862247�
A COUPLED JOINT EIGENVALUE DECOMPOSITION ALGORITHM FOR CANONICAL
POLYADIC DECOMPOSITION OF TENSORS
Rémi André, Xavier Luciani, and Eric Moreau,
Aix Marseille Université, CNRS, ENSAM, LSIS, UMR 7296, 13397 Marseille, France.
Université de Toulon, CNRS, LSIS, UMR 7296, 83957 La Garde, France.
ABSTRACT
In this paper we propose a novel algorithm to compute the joint eigenvalue decomposition of a set of squares matri-ces. This problem is at the heart of recent direct canonical polyadic decomposition algorithms. Contrary to the existing approaches the proposed algorithm can deal equally with real or complex-valued matrices without any modifications. The algorithm is based on the algebraic polar decomposition which allows to make the optimization step directly with com-plex parameters. Furthermore, both factorization matrices are estimated jointly. This “coupled” approach allows us to limit the numerical complexity of the algorithm. We then show with the help of numerical simulations that this approach is suitable for tensors canonical polyadic decomposition.
1. INTRODUCTION
Many iterative algorithms have been proposed to compute the Canonical Polyadic Decomposition (CPD) of tensors [1–4]. However these approaches can suffer from convergence prob-lems (local minima, slow convergence or high computational cost) and are very sensible to over-factoring and correlated factors [2,5]. In the meantime, a number of authors has shown how the CPD can be rewritten into a Joint EigenValue Decom-position problem (JEVD) [5–7].
JEVD consists in finding the eigenvector matrix A that jointly diagonalizes a given set of K non-defective matrices M(k)in the following way:
M(k)= AD(k)A−1, ∀k = 1, . . . , K. (1)
It is worth mentioning that in practice if the JEVD have a solution then it is unique up to a permutation and a scaling of the columns of A, that is assumed here. We refer to [6] for a sufficient and necessary condition of JEVD uniqueness. The link between CPD and JEVD has given birth to efficient CPD algorithms such as CFS [7], DIAG [5] or SECSI [8]. In-deed, although the JEVD is computed iteratively throughout a Jacobi-like procedure, this procedure involves small matri-ces and it usually requires very few iterations to converge, we thus speak of “direct” CPD algorithms. In comparison with iterative algorithms, direct algorithms are fast and less
sensible to convergence, over-factoring and correlated factors problems [5].
Several JEVD algorithms has been proposed in the literature. Most of them are based on Jacobi-like updates (sweeping pro-cedure) and look for a factorized form of the updating ma-trix. In [11] Fu and Gao introduced a first algorithm called sh-rt and based on the polar decomposition. The same fac-torization is also at the heart of JUST [12] and JDTM [5] al-gorithms while JET-U and JET-O resort to the LU factorisa-tion [10]. Finally in [13] we proposed an alternative approach, called JDTE that does not resort to any sweeping procedure. These algorithms have been extensively compared in [5], [10] and [13]. Most of them require significant modifications to deal with complex-valued matrices that make the optimiza-tion step more difficult during the sweeping procedure and somehow decrease the performances [10]. This is because complex versions of these algorithms need to estimate the real and the imaginary part or the modulus and the angle of the parameters separately. Therefore, in this paper we propose a versatile JEVD algorithm that can deal equally with real or complex-valued matrices, without any modifications. The method resorts to the algebraic polar decomposition for the factorization of the updating matrix. This allows us to make the optimization step directly upon the complex parameters. The second originality of the approach in the JEVD context is that both factorization matrices are estimated jointly. This “coupled” strategy limits the numerical complexity of the al-gorithm. It was originally introduced to deal with other Joint Diagonalization (JD) problems [14–16].
Notations Scalars are denoted by a lower case (a), vectors by a boldface lower case (a), matrices by a boldface upper case (A) and tensors by an upper case boldface calligraphic (A). ai is the i-th element of vector a, Ai,j is the (i, j)-th
element of matrix A and so on for tensors. Determinant and Frobenius norm of A are denoted det{A} and ||A|| respec-tively. |z| is the complex modulus of z.
2. MATRIX JEVD FOR TENSOR CPD
We recall here how the canonical polyadic decomposition can be rewritten into a JEVD problem. This algorithm called
DIAG has been introduced in [17] (under the name SALT) and deeply investigated in [5]. Another connection between both problems had been proposed in [7].
The rank-N CPD of a tensor T of order Q, size I1× · · · × IQ
is given by: Ti1,··· ,iQ= N X n=1 Xi(1) 1,n· · · X (Q) iQ,n, (2)
where X(q) defines the q-th factor matrix of size Iq × N .
Entries of T can be stored in an unfolding matrix T of size QP
p=1Ip×Q
Q
q=P +1Iqby regrouping, in an appropriate way,
the first P modes on the rows of T and the Q − P other modes on its columns. The CPD can then be rewritten in a matrix form:
T = Y(P,1)X Y(Q,P +1)X T
, (3)
with: Y(b,a)X = X(b) X(b−1) · · · X(a)
, (b > a) and where is the Khatri-Rao product. Note that one can easily deduce X(a), · · · , X(b)from Y(b,a)
X as explained in [5] so we
only recall here how to compute Y(P,1) X and Y
(Q,P +1)
X .
Let U SVHbe the singular value decomposition of T ,
trun-cated at order N , their exists a non singular square matrix A of size N × N such that:
Y(P,1)X = U A and Y(Q,P +1)X T = A−1SVH . (4) Y(Q,P +1) X T
can be seen as an horizontal block matrix:
Y(Q,P +1) X T =hφ(1)Y(Q−1,P +1) X T , · · · , φ(IQ)Y(Q−1,P +1) X Ti , (5)
where φ(1), · · · , φ(IQ) are the I
Q diagonal matrices built
from the IQrows of matrix X(Q). Then, (4) and (5) yield:
SVH= h Γ(1)T, · · · , Γ(IQ)Ti , (6) where Γ(k) = Y(Q−1,P +1)X φ(k)AT for k = 1, · · · , I Q.
As-suming that matrices Γ(k)and matrix Y(Q−1,P +1)
X are full
col-umn rank, then they all admit a Moore-Penrose matrix inverse denoted by ]. Thereby, we can define for any couple (k1, k2)
with k1= 1, · · · , IQ− 1 and k2= k1+ 1, · · · , IQ:
M(k1,k2) =def Γ(k1)]Γ(k2)T
, (7) = AD(k1,k2)
A−1, (8)
where D(k1,k2) = φ(k2)φ(k1)]are diagonal matrices. As a
result, A performs the JEVD of the set of matrices M and is estimated using a JEVD algorithm. Matrices Y(P,1)
X and
Y(Q,P +1)
X are then immediately deduced from A using (4).
3. A COUPLED JEVD ALGORITHM
We now describe the proposed coupled JEVD algorithm, that works equally in the real and complex case i.e matrices A, D(k) and M(k)can equally belong toR or C. Let us first recall the general scheme of JD algorithms. We want to build an estimate cA of matrix A (up to the scale and permuta-tion indeterminacy). At each iterapermuta-tion (also called sweep),
the algorithm computes a new set of N (N − 1)/2 nonsin-gular matrices, denoted B(i,j), with i = 1, · · · , N − 1 and j = i + 1, · · · , N and does the following updates of cA and M(k)(for each k) : c A ← cA N −1 Y i=1 N Y j=i+1 B(i,j) −1 M(k) ← N −1 Y i=1 N Y j=i+1 B(i,j)M(k) N −1 Y i=1 N Y j=i+1 B(i,j) −1. (9)
The process is repeated so that at the end all matrices M(k) are diagonals and cA = AΛΠ, where Λ and Π are diagonal and permutation matrices respectively. cA can be initialized using the identity matrix, single EVD of one matrix M(k) or generalized EVD. In practice, the updates in (9) consists in N (N − 1)/2 successive (i, j)-updates of cA and M(k), defined as: c A ← cA B(i,j)−1 M(k) ← B(i,j)M(k)B(i,j)−1. (10)
The underlying idea being that each (i, j)-update brings ma-trices M(k)closer to diagonal matrices by optimizing a “di-agonality” criterion.
We propose here a new way to build the (i, j)-updating matri-ces B(i,j). First, we define the (i, j)-updated matrices N(k) as: N(k)= B(i,j)M(k) B(i,j) −1 . (11)
We then look for the matrix B(i,j) that minimizes the cost function C [5, 10, 18], given by:
C(B(i,j)) = K X k=1 N (k) i,j 2 + N (k) j,i 2 . (12)
Thus we can impose the following structure for B(i,j):
∀ (p, q) /∈ {(i, i), (i, j), (j, i), (j, j)} , Bp,q(i,j)= δp,q,
B(i,j)i,i = b (i,j) 1 , B (i,j) i,j = b (i,j) 2 , B(i,j)j,i = b (i,j) 3 , B (i,j) j,j = b (i,j) 4 . (13)
where b(i,j)1 , b(i,j)2 , b(i,j)3 , b(i,j)4 are unknown parameters and δ is the Kronecker delta. Thanks to the scaling indeterminacy of the JEVD problem, we can also impose det{B(i,j)} = 1. Now, defining matrices: f M(k)= M (k) i,i M (k) i,j Mj,i(k) Mj,j(k) ! , fN(k)= N (k) i,i N (k) i,j Nj,i(k) Nj,j(k) ! , e B(i,j)= b (i,j) 1 b (i,j) 2 b(i,j)3 b (i,j) 4 ! , equation (11) becomes: f N(k)= eB(i,j)Mf (k) e B(i,j)−1 (14)
and the cost function becomes:
C( eB(i,j)) = K X k=1 Ne (k) 1,2 2 + Ne (k) 2,1 2 . (15)
In order to simplify the notations, in the following we will refer to B(i,j), eB(i,j), b(i,j)1 , b(i,j)2 , b(i,j)3 and b(i,j)4 as B, eB, b1, b2, b3and b4respectively.
We can immediately deduce from det{B} = 1 that:
e B−1= b4 −b2 −b3 b1 (16) and (14) yields: ( e N1,2(k) = b1b2(M (k) j,j − M (k) i,i ) + b 2 1M (k) i,j − b 2 2M (k) j,i, e N2,1(k) = −b3b4(M (k) j,j − M (k) i,i) − b 2 3M (k) i,j + b 2 4M (k) j,i. (17)
At this stage, one should note that during the sweeping proce-dure it is expected that the updating matrix B gets closer and closer to the identity matrix. Thus, if we assume that we are close enough to the solution we can reasonably assume that |b1| |b2| and |b4| |b3| and the previous system can be
simplified: ( e N1,2(k) ' b1b2(M (k) j,j − M (k) i,i ) + b 2 1M (k) i,j, e N2,1(k) ' −b3b4(M (k) j,j − M (k) i,i ) + b 2 4M (k) j,i. (18)
Now defining vectors b12 = [b1b2b21]T, b34= [b3b4b24]T and
matrices: M12= Mj,j(1)− Mi,i(1) Mi,j(1) . . . . . . Mj,j(K)− M(K) i,i M (K) i,j M21= Mi,i(1)− Mj,j(1) Mj,i(1) . . . . . . Mi,i(K)− M(K) j,j M (K) j,i yields: K X k=1 Ne (k) 1,2 2 = b12HM12HM12b12, K X k=1 Ne (k) 2,1 2 = b34HM21HM21b34. (19)
It then clearly appears that the two contributions to the cost function can be minimized separately and thus both parame-ters vectors b12 and b34can be computed independently. In
practice, we can first look for the 1-dimensional subspaces of b12and b34. In this purpose it is well known that the norm-1
vector that minimize f (x) = xHM
12HM12x is the
normal-ized eigenvector associated to the eigenvalue of M12HM12
of smallest modulus. In the same way the norm-1 vector that minimize g(x) = xHM
21HM21x is the normalized
eigen-vector associated to the eigenvalue of M21HM21of smallest
modulus. We denote e and f these two vectors and we have:
e = αb12,
f = βb34, (20)
where α and β are two unknown values. We now explain how we deduce eB from e and f by using the algebraic polar decomposition of eB.
Matrix polar decomposition states that any square real (com-plex) valued matrix X can be factorized as a product of an orthogonal (unitary) matrix and a positive-definite symmetric (hermitian) matrix. Actually, in the complex case we have
an alternative result stating that any square matrix X can be factorized as a product of a complex orthogonal matrix and a complex symmetric matrix if and only if XTX is similar
to XXT[19]. This factorization is called the algebraic polar
decomposition of X [20, 21]. A complex matrix Q is said complex orthogonal if QTQ = I and a complex matrix S is
said complex symmetric if ST
= S.
Therefore, let Q and S be the (complex) orthogonal and sym-metric matrices of the (algebraic) polar decomposition of eB i.e. eB = QS. Note that since eB is nonsingular, eBT
e B is similar to eB eBT and so the existence of the decomposition
is guaranteed. Because we have imposed det{ eB} = 1 and thanks to the scaling indeterminacy of the JEVD problem we can still impose det{Q} = det{S} = 1 and in these condi-tions there exists a couple of parameters (t, th) so that:
Q =√ 1 1 + t2 1 −t t 1 ; S = 1 p1 − t2 h 1 th th 1 . (21) Defining γ = √ 1 (1+t2)(1−t2 h) , eB = QS gives: b12= γ(1 − tth )(th− t) (1 − tth)2 ; b34= γ(t + th )(tth+ 1) (tth+ 1)2 . (22)
Now, (22) and (20) yield:
e1 e2 = th− t 1 − tth and f1 f2 = t + th tth+ 1 (23)
and finally we deduce:
( t2 h− 2ee21ff11+e+e21ff22th+ 1 = 0, t = e1−e2th the1−e2. (24)
Because thshould be close to 0, it is chosen as the root of the
binomial with the smallest modulus. From thwe immediately
deduce t then eB and B.
It is worth mentioning that: (i.) In the real case the algebraic polar decomposition is equivalent to the polar decomposition, therefore the proposed algorithm can be directly applied on real- or complex-valued matrices without any modification. (ii.) By construction, det{Q} = det{S} = det{ eB} = 1 so that existence of eB−1is guaranteed throughout the algorithm. Numerical complexity Numerical complexity of an itera-tive algorithm based on matrix computations can be approxi-mated by the number of multiplications computed during one iteration of the algorithm. Numerical complexities of JEVD algorithms are dominated by the updates of the matrix set. When dealing with real-valued matrices, the dominant term for the proposed algorithm is 4KN3. In comparison domi-nant terms for JUST, JDTM, sh-rt, JDTE, JET-O and JET-U algorithms are 11KN3, 8KN3, 8KN3, 2KN3, 2KN3and
KN3 respectively. Numerical complexities of complex ver-sions of the algorithms (when they exist) are still dominated by the updating step. Thus we can simply multiply the previ-ous values by 4. In conclusion, one iteration of the proposed
10 20 30 40 50 60 70 80 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 SNR (dB) log 10 (r X ) (median) DIAG−coupled DIAG−JDTM DIAG−JETU
(a)median value ofrXvs SNR
10 20 30 40 50 60 70 80 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 SNR (dB) log 10 (r X ) (average) DIAG−coupled DIAG−JDTM DIAG−JETU (b)average value ofrXvs SNR 10 20 30 40 50 60 70 80 105 106 107 108 SNR (dB) Γ (average) DIAG−coupled DIAG−JDTM DIAG−JETU (c)average value ofΓ vs SNR
Fig. 1.Comparison of DIAG-JDTM, DIAG-JETU and DIAG-coupled for the rank 5 CPD of complex valued-tensors of size 10 × 10 × 10.
algorithm is less costly than one iteration of other polar de-composition based algorithms but more costly than one iter-ation of JDTE or LU decomposition based algorithms. How-ever, the global (and practical) computational cost of each al-gorithm is obtained by multiplying its numerical complexity by the number of computed iterations before convergence and this number varies a lot from one algorithm to another.
4. NUMERICAL SIMULATIONS
Canonical polyadic decomposition of tensors is one of the main application of JEVD. Thereby, we chose to evaluate the performances of the proposed approach in this context by considering CPD of complex-valued tensors of order 3. In this purpose we compare 3 different implementations of the DIAG algorithm described in section 2; namely, DIAG-JDTM, DIAG-JETU and DIAG-coupled, respectively based on JDTM, JETU and the proposed algorithm. JDTM and JETU have been chosen as references because JDTM is very efficient to estimate the matrix of eigenvectors from noisy matrix sets while JETU has the lowest numerical complexity. The first comparison criterion, rX, is an average value of the
normalized mean squares error computed between actual and estimated factor matrices: rX = 13P
3 q=1
||X(q)
−Xd(q)||2 ||X(q)||2 ,
where dX(q) is the estimated factor matrix corrected from scaling and permutation indeterminacies. Algorithms perfor-mances are evaluated from 100 Monte-Carlo (MC) runs. For each run a new noise-free tensor is built using a new random draw of the 3 complex factor matrices. Then a Gaussian white noise is added in order to obtain the desired SNR value and the noisy tensor is decomposed using the three algorithms. When computing the DIAG algorithm, the JEVD is applied to the whole matrix set built from equation (7) (no matrix is removed). Matrix cA is initialized as the identity matrix. Dur-ing the JEVD process the squared off-diagonal components of the K matrices M(k)are summed and the obtained value is compared to the value computed at the previous sweep. The JEVD process is stopped when the relative deviation
between two successive values is smaller than 10−6or when a maximal number of 100 sweeps has been reached. Then the global computational cost of the JEVD step, Γ, is computed as defined previously. This gives us our second comparison criterion. Factor matrices used to build the tensors are of size 10 × 5, so rank-5 CPD of 10 × 10 × 10 tensors are performed and the JEVD problem is of size: K = 45, N = 5.
We present on figure 1 the evolution of the median and aver-age values of rX and of the average value of Γ with respect
to the SNR value. Regarding the estimation precision of the factor matrices, the proposed approach is the more efficient whatever the SNR value. Indeed DIAG-JDTM can compete only at the lowest SNR values (10 and 20 dB) whereas DIAG-JETU is clearly surpassed for the more realistic SNR values (this is particularly true below 40 dB). Furthermore, the av-erage value of rX shows a clear gap between the three
algo-rithms in favor of DIAG-coupled. Indeed during the JEVD process, JDTM and JET-U have not always converged con-trary to our coupled JEVD algorithm. Now regarding the global computational cost, figure 1(c) shows that the proposed algorithm is from 9.7 to 23.5 times less costly than JDTM (depending on the SNR value) and from 3.4 to 8 times more costly than JET-U. To sum up, DIAG-coupled appears here as the best choice in terms of estimation precision while being much less costly than the second more accurate solution.
5. CONCLUSION
In this paper we have introduced a new method for joint eigen-value decomposition. Contrary to classical approaches, this method works equally on real or complex-valued matrices by exploiting the algebraic polar decomposition and a coupled estimation of the factorization matrices. Numerical simula-tions demonstrate that it can be successfully used to compute the canonical polyadic decomposition of complex-valued ten-sors, rewritten as a JEVD problem. Indeed, in the represen-tative scenario that we have considered: the rank-5 CPD of a noisy tensor of size 10 × 10 × 10, the proposed algorithm al-lows a better estimation of the factors matrices than reference algorithms while keeping a low computational cost.
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